On the number of radial orderings of planar point sets

Let S be a set of n points in general position on the plane. For a given point p not in S, a radial ordering of S with respect to p is a circular ordering of the elements of S by angle around p. If S is bi-colored, with equal sized color classes, then a colored radial ordering of S with respect to p is a circular ordering by angle of the colors of the points of S around p, that is, permutations between points with the same color give the same ordering. The problem studied in this paper is to give bounds on the number of distinct non-colored and colored radial orderings of S. We show that the total number of distinct radial orderings of S is at most O(n) and at least Ω(n). In the bi-colored case we show that the number of distinct colored radial orderings of S is at most O(n) and at least Ω(n). An example of a point set with Θ(n) radial orderings is provided for both the non-colored and bi-colored case. In the bi-colored case, we also show how to generate examples with O(n) distinct colored radial orderings. Departamento Matemática Aplicada II, Universidad de Sevilla, [email protected], Partially supported by Grant MEC MTM2009-08625 Departamento de Matemáticas, Cinvestav, [email protected] Departamento Matemática Aplicada II, Universidad de Sevilla, [email protected], Partially supported by Grant MAEC-AECI and MEC MTM2009-08625